A072861 -id:A072861 - cp's OEIS frontend

A320059 Sum of divisors of n^2 that do not divide n.

0, 4, 9, 24, 25, 79, 49, 112, 108, 199, 121, 375, 169, 375, 379, 480, 289, 808, 361, 919, 709, 895, 529, 1591, 750, 1239, 1053, 1711, 841, 2749, 961, 1984, 1681, 2095, 1719, 3660, 1369, 2607, 2323, 3847, 1681, 5091, 1849, 4039, 3673, 3799, 2209, 6519, 2744, 5374

Offset: 1

Franklin T. Adams-Watters, Oct 04 2018

sigma(n^2) is always odd, so this sequence has the opposite parity from sigma(n): even if n is a square or twice a square, odd otherwise.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000203, A065764, A072861, A054785.

[DivisorSigma(1, n^2) - DivisorSigma(1, n): n in [1..70]]; // Vincenzo Librandi, Oct 05 2018

map(n -> numtheory:-sigma(n^2)-numtheory:-sigma(n), [$1..100]); # Robert Israel, Oct 04 2018

Table[DivisorSigma[1, n^2] - DivisorSigma[1, n], {n, 70}] (* Vincenzo Librandi, Oct 05 2018 *)

PARI
```
a(n) = sigma(n^2)-sigma(n)
```

from _future_ import division
from sympy import factorint
def A320059(n):
    c1, c2 = 1, 1
    for p, a in factorint(n).items():
        c1 *= (p**(2*a+1)-1)//(p-1)
        c2 *= (p**(a+1)-1)//(p-1)
    return c1-c2 # Chai Wah Wu, Oct 05 2018

a(n) = sigma(n^2) - sigma(n).
a(n) = A065764(n) - A000203(n).
a(n) = n^2 iff n is prime. - Altug Alkan, Oct 04 2018

A361147 a(n) = sigma(n)^3.

Original entry on oeis.org

1, 27, 64, 343, 216, 1728, 512, 3375, 2197, 5832, 1728, 21952, 2744, 13824, 13824, 29791, 5832, 59319, 8000, 74088, 32768, 46656, 13824, 216000, 29791, 74088, 64000, 175616, 27000, 373248, 32768, 250047, 110592, 157464, 110592, 753571, 54872, 216000, 175616

Offset: 1

Vaclav Kotesovec, Mar 02 2023

Cf. A000203, A024916, A072861, A072379, A361179.

Cf. A000005, A000578, A035116, A319089.

Mathematica
```
Table[DivisorSigma[1, n]^3, {n, 1, 50}]
```
PARI
```
a(n) = sigma(n)^3;
```

for(n=1, 100, print1(direuler(p=2, n, (1 + p*X*(2 + 2*p + p^2*X)) / ((1-X)*(1-p*X)*(1-p^2*X)*(1-p^3*X)))[n], ", "))

Multiplicative with a(p^e) = ((p^(e+1)-1)/(p-1))^3.
Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(s-2) * zeta(s-3) * Product_{primes p} (1 + 1/p^(2*s-3) + 2/p^(s-1) + 2/p^(s-2)).
Sum_{k=1..n} a(k) ~ c * Pi^6 * zeta(3) * n^4 / 2160, where c = Product_{primes p} (1 + 2/p^2 + 2/p^3 + 1/p^5) = 2.83598357433419286770442457158038489640898183...
a(n) = A000578(A000203(n)).

A066293 a(n) = A000203(n)^2 - A001157(n) = sigma(n)^2 - sigma_2(n).

Original entry on oeis.org

0, 4, 6, 28, 10, 94, 14, 140, 78, 194, 22, 574, 26, 326, 316, 620, 34, 1066, 38, 1218, 524, 686, 46, 2750, 310, 914, 780, 2086, 58, 3884, 62, 2604, 1084, 1466, 1004, 6370, 74, 1790, 1436, 5890, 82, 6716, 86, 4494, 3718, 2534, 94, 11966, 798, 5394, 2284

Offset: 1

Labos Elemer, Dec 12 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A001157, A002117, A072861.

a[n_] := DivisorSigma[1, n]^2 - DivisorSigma[2, n]; Array[a, 50] (* Amiram Eldar, Jul 31 2019 *)

a(n) = sigma(n)^2 - sigma(n, 2); \\ Michel Marcus, Mar 22 2020

For p prime, a(p) = 2p.
From Amiram Eldar, Mar 17 2024: (Start)
a(n) = A072861(n) - A001157(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(3)/2 = 0.601028451579... . (End)

A068484 Numbers k that divide phi(k)^2 + sigma(k)^2.

Original entry on oeis.org

1, 2, 10, 45, 65, 180, 212, 222, 369, 588, 810, 864, 1274, 1521, 1836, 2548, 2940, 3114, 3552, 4770, 5496, 5684, 6027, 6642, 8820, 9140, 10464, 10614, 13311, 14688, 15210, 20737, 21600, 22776, 26900, 27000, 27270, 28476, 28518, 42212, 42336

Offset: 1

Benoit Cloitre, Mar 10 2002

a(275) > 7*10^7. - G. C. Greubel, Oct 15 2018

G. C. Greubel, Table of n, a(n) for n = 1..274

Cf. A072861 (sigma(n)^2), A127473 (phi(n)^2).

Filtered([1..42500],n->(Phi(n)^2+Sigma(n)^2) mod n=0); # Muniru A Asiru, Oct 16 2018

with(numtheory): select(n->modp(phi(n)^2+sigma(n)^2,n)=0,[$1..42500]); # Muniru A Asiru, Oct 16 2018

Select[Range[7000], IntegerQ[(EulerPhi[#]^2 + DivisorSigma[1, #]^2)/#] &] (* G. C. Greubel, Oct 15 2018 *)

A109693 Decimal expansion of Sum_{k>=1} 1/sigma(k)^2.

Original entry on oeis.org

1, 3, 0, 6, 4, 5, 6, 5, 1, 2, 0, 3, 8, 9, 5, 0, 5, 6, 8, 0, 1, 0, 7, 4, 9, 4, 8, 7, 0, 9, 1, 2, 7, 1, 5, 4, 9, 7, 5, 8, 3, 9, 0, 7, 9, 1, 5, 6, 6, 4, 9, 1, 0, 3, 7, 3, 6, 0, 9, 6, 9, 9, 5, 9, 8, 6, 1, 5, 3, 4, 2, 6, 4, 5, 7, 6, 6, 8, 2, 8, 7, 1, 5, 9, 9, 8, 1

Offset: 1

Franklin T. Adams-Watters, Aug 07 2005

			1.3064565120...

Cf. A000203 (sigma function), A072861.

$MaxExtraPrecision = m = 1000; f[p_, m_] := 1 + Sum[(p - 1)^2/(p^(k + 1) - 1)^2, {k, 1, m}]; c = Rest[CoefficientList[Series[Log[f[1/x, m]], {x, 0, m}], x]]*Range[m]; RealDigits[f[2, Infinity] * Exp[NSum[Indexed[c, n]*((PrimeZetaP[n] - 1/2^n)/n), {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]] (* Amiram Eldar, Nov 14 2020 *)

my(N=1000000000); prodeuler(p=2,N, sum(k=1,200/log(p),if(k==1,1.,1./((p^k-1)/(p-1))^2)))*(1+1/N/log(N))

Product_{p prime} Sum_{k>=0} 1/sigma(p^k)^2.

More terms from Amiram Eldar, Nov 14 2020

A127574 Triangle T(n,k) = k*sigma(n) if k divides n, else 0.

Original entry on oeis.org

1, 3, 6, 4, 0, 12, 7, 14, 0, 28, 6, 0, 0, 0, 30, 12, 24, 36, 0, 0, 72, 8, 0, 0, 0, 0, 0, 56, 15, 30, 0, 60, 0, 0, 0, 120, 13, 0, 39, 0, 0, 0, 0, 0, 117, 18, 36, 0, 0, 90, 0, 0, 0, 0, 180, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 132, 28, 56, 84, 112, 0, 168, 0, 0, 0, 0, 0, 336

Offset: 1

Gary W. Adamson, Jan 19 2007

			First few rows of the triangle are:
   1;
   3,  6;
   4,  0, 12;
   7, 14,  0, 28;
   6,  0,  0,  0, 30;
  12, 24, 36,  0,  0, 72;
  ...

Cf. A127093, A127573, A064987, A000203, A072861 (row sums).

T(n,k) = Sum_{j=k..n} A130208(n,j)*A127093(j,k), product of the two infinite lower triangular matrices.
T(n,1) = A000203(n).
T(n,n) = A064987(n).

A180608 O.g.f.: exp( Sum_{n>=1} A067692(n)*x^n/n ), where A067692(n) = [sigma(n)^2 + sigma(n,2)]/2.

Original entry on oeis.org

1, 1, 4, 8, 21, 39, 93, 171, 364, 675, 1338, 2433, 4641, 8282, 15222, 26811, 47920, 83046, 145288, 248164, 425970, 718303, 1213106, 2020540, 3365352, 5541996, 9115640, 14856657, 24164430, 39002462, 62800603, 100454208, 160257140

Offset: 0

Paul D. Hanna, Oct 10 2010

nonn

sigma(n) = A000203(n), sum of divisors of n;

sigma(n,2) = A001157(n), sum of squares of divisors of n.

			O.g.f.: A(x) = 1 + x + 4*x^2 + 8*x^3 + 21*x^4 + 39*x^5 + 93*x^6 +...
log(A(x)) = x + 7*x^2/2 + 13*x^3/3 + 35*x^4/4 + 31*x^5/5 + 97*x^6/6 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A067692, A000203, A001157, A072861, A156302.

nmax = 40; $RecursionLimit -> Infinity; a[n_] := a[n] = If[n == 0, 1, Sum[(DivisorSigma[1, k]^2 + DivisorSigma[2, k])/2*a[n-k], {k, 1, n}]/n]; Table[a[n], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 28 2024 *)

{a(n)=polcoeff(exp(sum(m=1, n, (sigma(m)^2+sigma(m,2))/2*x^m/m)+x*O(x^n)), n)}

log(a(n)) ~ 3*(7*zeta(3))^(1/3) * n^(2/3) / 2^(4/3). - Vaclav Kotesovec, Oct 29 2024

A344347 Numbers k such that sigma(k)^2 is divisible by k-1.

Original entry on oeis.org

2, 3, 5, 10, 33, 55, 82, 129, 136, 145, 261, 351, 385, 406, 442, 513, 649, 897, 1090, 2241, 4726, 5185, 8650, 13601, 17101, 17641, 18241, 26625, 26937, 29697, 29953, 32896, 34561, 35841, 38417, 44955, 46081, 46593, 51985, 63505, 65703, 66249, 84376, 93313, 97903

Offset: 1

Zdenek Cervenka, May 15 2021

nonn

			For k=10, sigma(10)^2 / (10-1) = 18^2 / 9 = 324 / 9 = 36.

Cf. A072861, A263928.

Select[Range[2, 10^5], Divisible[DivisorSigma[1, #]^2, # - 1] &] (* Amiram Eldar, May 15 2021 *)

list(nn) = for(n=2, nn, if (sigma(n)^2 % (n-1) == 0, print1(n, ", ")))
list(100000)

A379635 Triangle read by rows: T(n,k) = A000203(k)*A000203(n-k+1), n >= 1, k >= 1.

Original entry on oeis.org

1, 3, 3, 4, 9, 4, 7, 12, 12, 7, 6, 21, 16, 21, 6, 12, 18, 28, 28, 18, 12, 8, 36, 24, 49, 24, 36, 8, 15, 24, 48, 42, 42, 48, 24, 15, 13, 45, 32, 84, 36, 84, 32, 45, 13, 18, 39, 60, 56, 72, 72, 56, 60, 39, 18, 12, 54, 52, 105, 48, 144, 48, 105, 52, 54, 12, 28, 36, 72, 91, 90, 96, 96, 90, 91, 72, 36, 28

Offset: 1

Omar E. Pol, Jan 14 2025

			Triangle begins:
   1;
   3,   3;
   4,   9,   4;
   7,  12,  12,   7;
   6,  21,  16,  21,   6;
  12,  18,  28,  28,  18,  12;
   8,  36,  24,  49,  24,  36,   8;
  15,  24,  48,  42,  42,  48,  24,  15;
  13,  45,  32,  84,  36,  84,  32,  45,  13;
  18,  39,  60,  56,  72,  72,  56,  60,  39,  18;
  12,  54,  52, 105,  48, 144,  48, 105,  52,  54,  12;
  28,  36,  72,  91,  90,  96,  96,  90,  91,  72,  36,  28;
  14,  84,  48, 126,  78, 180,  64, 180,  78, 126,  48,  84,  14;
  ...
For n = 10 the calculation of the row 10 is as follows:
    k    A000203         T(10,k)
    1       1   *  18   =   18
    2       3   *  13   =   39
    3       4   *  15   =   60
    4       7   *   8   =   56
    5       6   *  12   =   72
    6      12   *   6   =   72
    7       8   *   7   =   56
    8      15   *   4   =   60
    9      13   *   3   =   39
   10      18   *   1   =   18
                 A000203
.

Index entries for sequences related to sigma(n).

Column 1 and leading diagonal give A000203.
Middle diagonal gives A072861.
Row sums give A000385.
Cf. A221529.

T[n_,k_]:=DivisorSigma[1,k]*DivisorSigma[1,n-k+1];Table[T[n,k],{n,12},{k,n }]//Flatten (* James C. McMahon, Jan 15 2025 *)

PARI
```
T(n, k)=sigma(k)*sigma(n-k+1)
```

A383614 The unique sequence such that Sum_{d|n} d*a(d)^(n/d) = sigma(n)^2 for every n.

Original entry on oeis.org

1, 4, 5, 4, 7, -10, 9, -44, -23, -197, 13, -845, 15, -2340, -701, -9164, 19, -31578, 21, -124979, -11355, -381326, 25, -1778580, -3323, -5162265, -212899, -21915630, 31, -70256029, 33, -311369996, -4439583, -1010580635, -129393, -4135827284, 39, -14467258386

Offset: 1

Yifan Xie, May 02 2025

Replace the sequence A072861 on the right-hand side of the equation with any integer sequence. It can be proved that the resulting sequence {a(n)} contain only integer terms if and only if for any prime p and positive integer n such that val(n, p) = k, p^k divides s(n) - s(n/p). The simplest sequence satisfying this property is A000203, and the resulting sequence {a(n)} is the constant sequence of 1's.

In 2025 China Team Selection Test, Test 4, Day 1, Problem 1, this sequence gives {z(n)} when {x(n)} and {y(n)} are constant sequences of 1's.

			For n = 1, the equation gives a(1) = sigma(1)^2 = 1;
For n = 6, the equation gives 1*1^6 + 2*4^3 + 3*5^2 + 6*a(6) = sigma(6)^2 = 144, so a(6) = -10.

Art of Problem Solving, 2025 China Team Selection Test, Test 4, Day 1, Problem 1

Cf. A000203, A072861.

lista(nn) = {my(v=vector(nn)); v[1] = 1; for(n=2, nn, s = sigma(n)^2; fordiv(n, d, s -= d*v[d]^(n/d)); v[n]=s/n); v}

For prime p, a(p) = p + 2.

cp's OEIS Frontend

A320059 Sum of divisors of n^2 that do not divide n.

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Magma

Maple

Mathematica

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Formula

A361147 a(n) = sigma(n)^3.

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Mathematica

PARI

PARI

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A066293 a(n) = A000203(n)^2 - A001157(n) = sigma(n)^2 - sigma_2(n).

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Mathematica

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A068484 Numbers k that divide phi(k)^2 + sigma(k)^2.

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GAP

Maple

Mathematica

A109693 Decimal expansion of Sum_{k>=1} 1/sigma(k)^2.

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Extensions

A127574 Triangle T(n,k) = k*sigma(n) if k divides n, else 0.

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A180608 O.g.f.: exp( Sum_{n>=1} A067692(n)*x^n/n ), where A067692(n) = [sigma(n)^2 + sigma(n,2)]/2.

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Mathematica

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Formula

A344347 Numbers k such that sigma(k)^2 is divisible by k-1.

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